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The intersection number for forcing notions

Andrés F. Uribe-Zapata

Abstract

Based on works of Saharon Shelah, Jakob Kellner, and Anda Tănasie for controlling the cardinal characteristics of the continuum in ccc forcing extensions, in the author's master's thesis was introduced a new combinatorial notion: the intersection number for forcing notions, which was used in such thesis to build a general theory of iterated forcing using finitely additive measures. In this paper, we present the definition of such a notion and prove some of its fundamental properties in detail. Additionally, we introduce a new linkedness property called $μ$-intersection-linked, prove some of its basic properties, and provide some interesting examples.

The intersection number for forcing notions

Abstract

Based on works of Saharon Shelah, Jakob Kellner, and Anda Tănasie for controlling the cardinal characteristics of the continuum in ccc forcing extensions, in the author's master's thesis was introduced a new combinatorial notion: the intersection number for forcing notions, which was used in such thesis to build a general theory of iterated forcing using finitely additive measures. In this paper, we present the definition of such a notion and prove some of its fundamental properties in detail. Additionally, we introduce a new linkedness property called -intersection-linked, prove some of its basic properties, and provide some interesting examples.
Paper Structure (7 sections, 23 theorems, 15 equations)

This paper contains 7 sections, 23 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finitely additive measures on Boolean algebras
  4. Trees
  5. Forcing notions
  6. The intersection number for forcing notions
  7. $\mu$-intersection-linkedness

Key Result

Lemma 2.4

Let $\mathscr{B},$$\mathscr{C}$ be Boolean algebras, $\mu$ a strictly positive finitely additive probability measure on $\mathscr{B}$ and $\theta$ an infinite cardinal. Assume that $\mathscr{C}$ is a Boolean subalgebra of $\mathscr{B}$ and $\mu$ has the $\theta$-density property witnessed by $S.$ If

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 37 more